tl;dr: ignorance can be optimal

Consider a two-alternative choice test:

Which city has a larger population: City A or City B?

City A and City B are selected at random from a set of $N$ cities. Of these cities, $n$ are recognizable. The meaning of recognizable is central to the argument in the sense that recognition depends on context: familiarity with Paris (France) does not necessarily imply useful recognition of Paris (Ontario).

The expected proportion of correct inferences is

$$ f(n) = 2 \left( \frac{n}{N} \right) \left( \frac{N-n}{N-1} \right) \alpha + \left( \frac{N-n}{N} \right) \left( \frac{N-n-1}{N-1} \right) \left( \frac{1}{2} \right) + \left( \frac{n}{N} \right) \left( \frac{n-1}{N-1} \right) \beta. $$

Definitions:

1. Recognition validity ($\alpha$) is the probability of a correct answer when only one choice is recognized, describing how well recognition correlates with the criterion.
2. Knowledge validity ($\beta$) is the probability of a correct answer when both choices are recognized, describing how well a choice is supported by available knowledge.

Reference

Gigerenzer, Gerd, and Peter M. Todd. Simple Heuristics That Make Us Smart. Oxford University Press, 1999.